Question

Find the exact value of the trigonometric function given that sin u = 8/17 and cos v = -3/5 (both u and v are in Quadrant II) cot (u + v)

Answer 1

The given values are sin u = 8/17 and cos v = -3/5

Apply pythagoren identity formula : cos² u = 1 - sin²u

= 1 - (8/17)²

cos² u  =  1 - 64/289

cos u = 15/17

Apply pythagoren identity formula: sin²v  = 1- cos² v

= 1- (-3/5)²

sin²v  = 1- 9/25

sin v = 4/5

cot(u + v) = 1/ tan(u +v)

Formula : tan(u +v) = (tan u + tan v) / [1 - (tan u)(tanv)]

cot(u + v) = [1 - (tan u)(tanv)] / (tan u + tan v)

tan u = sin u/cosu  = 8/15

tan v = sin v/cosv  = -4/3

cot(u + v) =[1 - (tan u)(tanv)] / (tan u + tan v)

= [1 - (8/15)(-4/3)] / ( 8/15 - 4/3)

= [1 + (32/45)] / (-12/15)

= [77/45] / (-12/15)

= - 77/36

cot (u + v)  = - 77/36

Comments

cot(u+v) = - 13/84.

Answer 2

Calculate the cot(u + v) by using the sum formula : cot(u + v) = [cot(u) * cot(v) - 1] / [cot(u) + cot(v)].

sin(u) = 8/17 and cos(v) = -3/5 and both angles u and v are lies in second quadrant.

By using the Pythagorean identity sin² θ + cos² θ = 1, to obtain

(8/17)² + cos² u = 1

cos² u = 1 - (8/17)² = (289 - 64)/289 = 225/289

cos u = ± √(225/289) = ± 15/17

Since cos θ is negative in second quadrant, to obtain cos u = - 15/17.

Quotient Identity : cot u = cos(u)/sin(u) = (- 15/17)/(8/17) = - 15/8.

 

By using the Pythagorean identity sin² θ + cos² θ = 1, to obtain

sin² v + (-3/5)² = 1

sin² v = 1 - (-3/5)² = (25 - 9)/25 = 16/25.

sin v = ± √(16/25) = ± 4/5

Since sin θ is positive in second quadrant, to obtain sin v = 4/5.

Quotient Identity : cot v = cos(v)/sin(v) = (-3/5)/(4/5) = - 3/4.

 

cot(u+v) = [cot(u)*cot(v) - 1] / [cot(u) + cot(v)]

               = [(-15/8)*(-3/4) - 1] / [(-15/8) + (-3/4)]

               = [(45/32) - 1] / [ - 15/8 - 3/4 ]

               = [(45 - 32)/32] / [(-15-6)/8]

               = [13/32] / [-21/8]

               = (13)(8) / (32)(-21)

cot(u+v) = - 13/84.